Probability that sum of two random variables is greater than 1

Homework Statement

Let us choose at random a point from the interval (0,1) and let the random variable [itex]X_1[/itex] be equal to the number which corresponds to that point. Then choose a point at random from the interval (0,[itex]x_1[/itex]), where [itex]x_1[/itex] is the experimental value of [itex]X_1[/itex]; and let the random variable [itex]X_2[/itex] be equal to the number which corresponds to this point.

Homework Statement

Let us choose at random a point from the interval (0,1) and let the random variable [itex]X_1[/itex] be equal to the number which corresponds to that point. Then choose a point at random from the interval (0,[itex]x_1[/itex]), where [itex]x_1[/itex] is the experimental value of [itex]X_1[/itex]; and let the random variable [itex]X_2[/itex] be equal to the number which corresponds to this point.

The Attempt at a Solution

As has been noted, that integral does not evaluate to zero. (It evaluates to one.)

Much more importantly, that is the wrong integral. Look what happens when x2 is, for example, 3/4. The integration limits for x1 are 1/4 to 1. When x1 is 1/4, the maximum possible value for x2 is 1/4. An x2 value of 3/4 is not possible.

Bottom line: Your integral covers more than the sample space.

If you want to use this approach, I suggest drawing a picture. You want the portion of the sample space for which x1+x2≥1.